Reductionist modeling of calcium-dependent dynamics in recurrent neural networks

Mathematical analysis of biological neural networks, specifically inhibitory networks with all-to-all connections, is challenging due to their complexity and non-linearity. In examining the dynamics of individual neurons, many fast currents are involved solely in spike generation, while slower currents play a significant role in shaping a neuron's behavior. We propose a discrete map approach to analyze the behavior of inhibitory neurons that exhibit bursting modulated by slow calcium currents, leveraging the time-scale differences among neural currents. This discrete map tracks the number of spikes per burst for individual neurons. We compared the map's predictions for the number of spikes per burst and the long-term system behavior to data obtained from the continuous system. Our findings demonstrate that the discrete map can accurately predict the canonical behavioral signatures of bursting performance observed in the continuous system. Specifically, we show that the proposed map a) accounts for the dependence of the number of spikes per burst on initial calcium levels, b) explains the roles of individual currents in shaping the system's behavior, and c) can be explicitly analyzed to determine fixed points and assess their stability.